For the investigation of the effect of spacetime curvature and arbitrary field coupling on symmetry restoration, we have computed by zeta-function regularization the one-loop effective potential for a self-interacting scalar field arbitrarily coupled to a static Robertson-Walker universe. The classical contribution of curvature can restore broken symmetries for a wide range of parameters from conformal to near-minimal couplings whereas the quantum contribution is important only for a small range of values near the minimal coupling point. The symmetry behavior of the two extremes is found to be very different.